A symmetric random walk defined by the time-one map of a geodesic flow

In this note we consider a symmetric Random Walk defined by a \begin{document}$ (f, f^{-1}) $\end{document} Kalikow type system, where \begin{document}$ f $\end{document} is the time-one map of the geodesic flow corresponding to an hyperbolic manifold. We provide necessary and sufficient conditions for the existence of an stationary measure for the walk that is equivalent to the volume in the corresponding unit tangent bundle. Some dynamical consequences for the Random Walk are deduced in these cases.